I turn all of my bottle stoppers on a pin chuck... I have a 3/8" rod that I've cut to length, and fitted into a collet in my Beals collet.... then filed a flat on one side and cut a small nail to fit on the flat... drill a 3/8" hole in the blank to the depth of the flat, push it on and twist the blank until the nail catches... no problems with in stability, the bottom of the stopper is away from the mandrel and available to turn to shape. I do suggest cutting several nails... I usually lose one per session... I can turn 3-4 stoppers per hour using this method..... I do have to epoxy the stopper in place, but have always done that anyway.
This method works for both the stainless steel and the silicone stoppers on the maple dowels... I do have to do a quick spin on my belt sander for the dowels to fit in the hole sometimes.